Since the automorphisms of \({D^{\mathrm {IV}}_3}\)  act transitively on \({\mathcal {R}}\) , we may assume that \(H(p) = p = (0,\frac{1}{2},\frac{i}{2})\) . The rational map \(\Phi \)
\(\Phi (z,\zeta ,w)=\left(\frac{2i z}{2i + w}, \ \frac{2i - w - 2i \zeta -(w\zeta + iz^2)}{2(2i+w)},\ \frac{i\left(2i - w + 2i \zeta +(w\zeta + iz^2) \right)}{2(2i+w)}\right)\) 
sends a neighbourhood \(V\)  of the origin biholomorphically onto some neighbourhood \(U\)  of \(p\)  with \(\Phi (0,0,0) = p\)  and locally \(\mathcal {X}\)  into \({\mathcal {R}}\) . Then \(\gamma :=\Phi ^{-1} \circ H\circ \Phi \)  defines a local CR diffeomorphism of \(\mathcal {X}\)  fixing the origin, i.e., \(\gamma \)  represents an element of the stability group \(\operatorname{Aut}(\mathcal {X},0)\)  and has the form (REF ) for an appropriate quadruple of parameters \(a, u, t\) , and \(\lambda \) . Therefore, \(H = \Phi \circ \gamma \circ \Phi ^{-1}\)  agrees with a rational automorphism (in fact, an isometry) of \({D^{\mathrm {IV}}_3}\)  as can be checked directly. Indeed, explicitly one can take
\(\Phi ^{-1}(z_1,z_2,z_3)=\left(\frac{2z_1}{1+z_2 -i z_3},\ -\frac{z_2 + i z_3 + z_1^2+z_2^2+z_3^2}{1+z_2 -i z_3}, \ \frac{2i(1-z_2 + i z_3)}{1+z_2 -i z_3}\right)\) 

Then \(\Phi \circ \gamma \circ \Phi ^{-1}\)  is a second degree rational mapping of the form
\(\widetilde{H}=\Phi \circ \gamma \circ \Phi ^{-1}=\left(\frac{\psi _1}{\Delta },\ \frac{\psi _2}{2\Delta },\ \frac{\psi _3}{2\Delta }\right)\) 
where the components of the map are
\(\psi _1 & = 2u\lambda (ia + z_1 + 2z_2 \operatorname{Im}(a) -z_3 \operatorname{Re}(a) -i \bar{a} (z_1^2 + z_2^2+z_3^2) ), \\\psi _2 & = 1+2|a|^2 -\lambda ^2 -2it -2a^2 u^2 + 4iz_1 \left(a u^2-\bar{a}\right)\\& \quad + z_2\left( u^2 \left(2 a^2-2 | a|^2 +\lambda ^2-2 i t+1\right)+1-2| a|^2 +\lambda ^2+2 \bar{a}^2+2 i t\right)\\& \quad -i z_3\left(u^2 \left(2 a^2+2| a|^2-\lambda ^2+2 i t-1\right)+1-2| a|^2+\lambda ^2-2 \bar{a}^2+2 i t\right)\\& \quad + \left(u^2 \left(1+2 |a|^2-\lambda ^2+2 i t\right)-2 \bar{a}^2\right)(z_1^2+z_2^2+z_3^2),\\\psi _3 & = i\bigl (1+2 |a|^2-\lambda ^2-2 i t + 2 a^2 u^2 - 4i z_1 (au^2 + \bar{a}) \\& \quad + z_2(1-2 |a|^2+\lambda ^2+2 \bar{a}^2+2 i t - u^2(1-2|a|^2 +\lambda ^2 + 2a^2 -2i t)) \\& \quad -i z_3\left(u^2 \left(1-2 |a|^2+\lambda ^2-2 a^2-2 i t\right)+1-2|a|^2+\lambda ^2-2 \bar{a}^2+2 i t\right)\\& \quad -\left(u^2 \left(1+2 |a|^2-\lambda ^2+2 i t+1\right)+2 \bar{a}^2\right)(z_1^2+z_2^2+z_3^2)\bigr ),\) 

and the common denominator is
\(\Delta =(1+2 |a|^2+\lambda ^2-2 i t)-2\bar{a}^2(z_1^2+z_2^2+z_3^2) - 4i \bar{a}z_1\\+ (1-2 |a|^2- \lambda ^2 + 2 \bar{a}^2 + 2 i t)z_2-i(1-2 |a|^2 -\lambda ^2 -2 \bar{a}^2 + 2 i t) z_3.\) 

Equation (REF ) gives the most general parametric formula for the CR automorphisms of the germ \(({\mathcal {R}},p)\) .
To show that the rational map \(\widetilde{H}\)  restricts to an automorphism of \({D^{\mathrm {IV}}_3}\) , it suffices to show that \(\Delta \)  does not vanish on \({D^{\mathrm {IV}}_3}\) .
If we write
\(W = \left(\frac{-2ia}{1+2|a|^2+\lambda ^2+2it},\ \frac{\lambda ^2-1+2|a|^2-2a^2+2it}{2(1+2|a|^2+\lambda ^2+2it)},\ \frac{i(\lambda ^2-1+2|a|^2+2a^2 +2it)}{2(1+2|a|^2+\lambda ^2+2it)}\right)\) 
then \(W\in {D^{\mathrm {IV}}_3}\)  and \(\widetilde{H}(W) =(0,0,0)\)  and further more
\(\Delta =(1+2 |a|^2+\lambda ^2-2 i t)\left(1 - 2 \overline{W}^t Z + \overline{(W^t W)}\,(Z^t Z)\right).\) 
But from the explicit formula for the Bergman kernel \(K_{{D^{\mathrm {IV}}_3}}(Z,W)\)  of \({D^{\mathrm {IV}}_3}\)  (see [1]}), we have
\(K_{{D^{\mathrm {IV}}_3}}(Z,W) = \mathrm {volume}({D^{\mathrm {IV}}_3})^{-1} \left(\frac{\Delta }{1+2 |a|^2+\lambda ^2-2 i t}\right)^{-3}\) 
and the nonvanishing of \(\Delta \)  on \({D^{\mathrm {IV}}_3}\)  is equivalent to the well-known fact that \({D^{\mathrm {IV}}_3}\) , being a homogeneous complete circular domain, has nonvanishing Bergman kernel (i.e, is a Lu-Qi Keng domain).
